1. Introduction

Electronics circuits keep shrinking in dimensions, according to famed Moore’s law, with FET gate lengths in the laboratory being in the tens of nm range. In contrast, photonic circuit elements and waveguides have lateral dimensions on the order of the wavelength, and the circuit elements normally are tens to thousands of wavelengths in length. A key to make integrated photonics more low-cost and dramatically more useful technology is a drastic reduction of size. To build highly integrated planar lightwave circuits, many efforts have been made, and several waveguide structures have been introduced, such as the waveguides based on high-index-contrast wires [1, 2], photonic crystals [3], and surface plasmons (SPs) [4–15]. Among these waveguide structures, SP waveguides, which utilize the fact that light can be confined in a single interface between a metal and dielectric [4], can offer the best confinement for the light field. Long-range low-loss SP waveguides based on thin metal films or stripes have been intensively investigated both theoretically and experimentally [5, 6]. However, this type of long-range SP waveguide does not have the potential for high integration, since the light field widely extends into the surrounding dielectric media [7]. By using metal claddings, the light field can be tightly confined. Some classical structures for wave guiding in microwave region (e.g. metallic stripe lines) have been intensively studied [8]. However, the response function of a metal in optical region is vastly different from that in microwave region, and some calculation models and approximations used in microwave region for analyzing the metallic structures may be not valid in optical region. Thus, it is necessary to study SP waveguides in the optical region with some accurate methods. Some three-dimensional metallic structures which can support highly localized fields at optical frequencies have been analyzed, e.g. a rectangular dielectric waveguide between two metallic sheets [9], two corrugated metallic sheets [10, 11] and V-grooves in metals [12–15].

In this paper, a novel three-dimensional SP waveguide structure is proposed for high integration. The finite-difference mode solver [16] and the finite-difference time-domain (FDTD) method [17] are used to study the guided modes and the light propagation in the proposed SP waveguide structure. Some basic requirements for high integration (e.g. coupling between two closely positioned waveguides and 90° bending) are studied. The proposed SP waveguide structure offers a high confinement for the light field, and is easier to fabricate as compared with other types of SP waveguide structures for high integration [9, 11, 14].

 

Fig. 1. (a) Schematic diagram (cross-sectional x-y plane) and (b) a fabricated structure (x-z plane view) of the proposed SP waveguide. Amplitude distributions of the fundamental mode when (c) w=50nm, h=200nm and (d) w=50nm, h=60nm. The dotted lines in (c) and (d) indicate the boundary of different materials.

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2. Waveguide structures and mode properties

The cross-section of the proposed SP waveguide structure is shown in Fig. 1(a). The working wavelength is set to be λ ₀=632.8nm. Silica (with refractive index of 1.47) is employed as a substrate. A thin film of metal (assume to be silver with refractive index of 0.119+3.964j [18]) is deposited on the substrate, and a small trench is etched down to form the waveguide. The whole structure is then coated with a polymer (assume to be PMMA with refractive index of 1.49) for matching (roughly) the refractive index of the substrate (we will show later that this upper-cladding is crucial in making the proposed SP waveguide work) and protection. In the following simulations, we assume the polymer upper-cladding is thick enough and the air above is not taken into consideration. A TE polarized light (i.e., the electrical field is mainly along the x direction) can be confined in the small trench. The lateral confinement is achieved through the metal film with the SP wave. The vertical confinement is achieved through the index guiding since the effective index of the SP wave supported in the trench is higher than the refractive indices of the substrate and upper-cladding. Figure 1(b) shows a picture of a fabricated SP waveguide structure before coating the polymer. The silica substrate and the metal film are deposited on a wafer with the PECVD (Plasma Enhanced Chemical Vapor Deposition) and sputtering technology, respectively. The trenches (about 90nm wide) are patterned with the E-beam lithography. The metal is anisotropically etched in an RIE (Reactive Ion Etcher). Finally, a polymer layer can be spin-coated on the whole structure.

 

Fig. 2. Real part β_r of the propagation constant for different modes when w=50nm. The insets show some typical mode profiles of E_x for the corresponding modes. The dot-dashed line indicates the real part β_sp of the propagation constant for the SP wave supported by the metal film without the trench.

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The mode properties are studied with the finite-difference mode solver [16]. We set the discretization lattices to be 5nm and 10nm in the x and y directions, respectively. The whole computational domain has a size of 1.5μm×1.5μm, which is large enough to obtain accurate propagation constants. Figure 1(c) and 1(d) show the typical profile of the fundamental mode in the proposed SP waveguide structure. It is worthwhile to note that the light field is highly confined in the lateral (x) direction, but may extend to a relatively large distance in the vertical (y) direction. This is related to the guiding mechanism (discussed above) of the proposed structure. Such a waveguide structure is suitable for highly integrated planar lightwave circuits, for which the lateral confinement is much more important than the vertical confinement. If the metal film is too thick, the proposed waveguide structure will support multimodes in the vertical direction. Figure 3 shows the real part β_r of the propagation constant and some typical mode profiles for different modes (the width of the trench is fixed at w=50nm). In this case, at least three guided modes exist when the thickness of the metal film reaches 500nm. Usually, the higher order modes have larger loss (i.e., larger imaginary part of the propagation constant). Furthermore, we cannot find any anti-symmetry SP mode (high order mode in the lateral dimension). This is because the narrow width of the trench is less than the cut-off width for the anti-symmetry SP mode. Note that the SP wave supported by the metal film without the trench [4] (the real part β_sp of the propagation constant for this SP wave is shown by the dot-dashed line in Fig. 2) is very helpful in analyzing the mode properties in the proposed structure. When β_r for a mode is larger than β_sp, the modal field is tightly confined in the trench (a guided mode; cf. Fig. 1(c)). When β_r is approaching β_sp, the field spreads more into the upper-cladding and substrate (cf. Fig. 1(d)). If β_r is less than β_sp, the field will not be confined in the trench any more and leak to the side in the lateral direction (i.e., a leaky mode; here a leaky mode usually has a larger loss as compared with a guided mode). Therefore, we can use the value of β_sp to separate well-confined guided modes from widely-spread leaky modes for the proposed SP waveguide. [Unlike for a conventional dielectric waveguide, we cannot distinguish between a leaky mode and a guided mode by whether the mode is lossy in the present case, since both of them are lossy here (due to the lossy metal).]

 

Fig. 3. Dependence of the mode properties on the structural parameters in the proposed SP waveguide. The two black solid lines indicate the boundaries of different regions. The colored curves give the contours of the propagation length (indicated with numbers) for the fundamental mode.

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Fig. 4. Real part β_r of the propagation constant for the fundamental mode as the refractive index of the cladding decreases when h=400nm. The dot-dashed line indicates the real part β_sp of the propagation constant for the SP wave supported by the metal film without the trench.

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Figure 3 illustrates the dependence of mode properties on the structural parameters of the proposed SP waveguide. The two black solid lines indicate the boundaries of the multi-mode region, single mode region and the region of no guided mode. The contours in Fig. 3 indicate the propagation length (defined as the propagation distance through which the amplitude of the field attenuates to 1/e; a smaller propagation length means a large loss of the guided mode) for the fundamental mode. From this figure one sees that the loss of the fundamental mode is smaller for a thicker metal film and a wider trench. Although loss is inevitable in an SP waveguide, we show in the next two sections that some critical dimensions of a planar lightwave circuit (e.g. the distance that two waveguides can be closely packed without a significant crosstalk, and the radius for a low loss bending) based on the proposed SP waveguides can be much smaller than the propagation length. This will give the possibility for building highly integrated circuits of the present SP waveguides within the range of several propagation lengths.

Here we would like to discuss the necessity of using a polymer upper-cladding (which has a similar refractive index as the silica substrate) in the proposed SP waveguide. Since most of the light field is confined in the trench, the refractive index there (same as the refractive index of the upper-cladding) will affect significantly on the propagation constant. If the refractive index of the upper-cladding decreases, β_r for the fundamental mode will decrease fast to β_sp (almost linearly; see Fig. 4). This results in the termination of any guided mode when the index of the upper-cladding is too low. For a fixed refractive index of the upper-cladding, decreasing the trench width or increasing the metal film thickness can help to increase β_r (cf. Figs. 2 and 4). In particular, to make the proposed SP waveguide structure work (i.e., support a guided mode) for air cladding (i.e., without polymer coating), we find that the width w of the trench should be smaller than 30nm and the aspect ratio (h/w) should be at least 15:1 (an aspect ratio of 1:1 suffices when a PMMA upper-cladding exists; cf. Fig. 3). This will make the fabrication difficult or impossible. Therefore, the upper-cladding is crucial in making the proposed SP waveguide work.

 

Fig. 5. (a) Two closely positioned SP waveguides. (b) The dependence of the coupling length on separation D when w=50nm and h=200nm. (c) The dependence of the coupling length on the metal film thicknesses h and the trench width w when D=150nm.

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3. Coupling between two closely positioned waveguides

To check the feasibility of the proposed SP waveguide for high integration, here we study the coupling between two parallel waveguides close to each other (see Fig. 5(a)). Figure 5(b) shows the coupling length between two SP waveguides when the structural parameters are chosen as w=50nm and h=200nm (which is in the single mode region; the propagation length is 2.86μm in this case; cf. Fig. 3). The coupling length is determined by the beat length of the two super modes (calculated with the finite-difference mode solver [16]) supported by the system of the two SP waveguides. A larger coupling length means the two waveguides have a smaller crosstalk. Even the separation is quite small (e.g. 150nm; one order smaller than the propagation length), the coupling length is still very large (e.g. 15μm; nearly one order larger than the propagation length). This indicates that the proposed SP waveguides could be densely packed without a noticeable crosstalk in the range of a few propagation lengths. Figure 5(c) shows the dependence of the coupling length on the structural parameters. From this figure one sees that a thicker metal film and a narrower trench are desirable for higher integration. Together with the consideration of the propagation length (analyzed in the above section; cf. Fig. 3), a thicker metal film is always preferred to obtain both larger propagation length and higher integration density. When choosing the width of the trench, we should make a compromise between the loss (wider trench is preferred) and integration density (narrower trench is preferred).

4. 90° bending

A 90° bending of the proposed SP waveguide (see Fig. 6(a)) is studied with the three-dimensional FDTD method with the perfectly matched layer as the boundary condition [17]. The discretization lattices in the x, y and z directions are 5nm, 10nm and 5nm, respectively. The whole computational domain has a size of 1.0μm×0.6μm×1.0μm. The structural parameters are again chosen as w=50nm and h=200nm. The power transmissivity through the 90° bending, the reflectivity, and the out-of-plane loss are shown in Fig. 6(d) as the bending radius increases. When the bending radius r is small, the loss is mainly caused by the reflection (see Fig. 6(b)). Increasing the bending radius can reduce the reflection loss. The out-of-plane loss, which is mainly caused by the mismatch of the mode profiles of the straight section and the bending section, also decreases as the bending radius increases. When the bending radius r is large, the loss of this structure is mainly caused by the material loss (see Fig. 6(c)). Thus, unlike a conventional dielectric waveguide, the power transmissivity through a 90° bending structure decreases as the bending radius increases further (since the length of the bending section increases). The transimissivity reaches a maximum of 77.2% when r=20nm. If we want to minimize the reflection, the bending radius r should be at least 100nm (the transimissivity is 74.8% when r=100nm). We can conclude that a sharp bending can be achieved with the proposed SP waveguide, and the bending loss is still acceptable for practical applications.

 

Fig. 6. (a) Schematic diagram of a 90° bending (top-viewing x-z plane). The amplitude distribution of H_y field in the x-z plane at the central depth of the metal films when the bending radius (b) r=0nm and (c) r=100nm. (d) The transimissivity, reflectivity and out-of-plane loss as the bending radius r increases. The transimissivity (reflectivity) is evaluated as the power propagating (reflected) through the output (input) facet of the bending section. The out-of-plane loss is evaluated as the power radiated out of the bending plane (the x-z plane). All these powers are normalized by the input power at the input facet of the bending section.

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5. Conclusion

To improve the integration density of planar lightwave circuits, we have introduced a novel SP waveguide which is formed by etching a small trench through a metallic thin film on a silica substrate. We have shown that an upper-cladding with a refractive index close to that of silica substrate is crucial to make the proposed SP waveguide work. We have also studied the coupling between two closely positioned waveguides and a 90° bending based on this SP waveguide numerically. Loss is inevitable in a tightly confined SP waveguide. However, this kind of highly confined SP waveguide is not used for transporting light energy on a longer path, but for constructing highly-integrated waveguide circuit components. Through the analysis of the coupling length and bending loss, we have shown that a reasonable separation distance (that two waveguides can be put without much coupling) and a reasonable bending radius (with acceptable bending loss) can be tens of times smaller than the propagation length. Therefore, some highly integrated circuit components based on the proposed SP waveguide can be fabricated within the propagation length (typically several microns). Thus, it is possible to make some highly integrated circuits (based on the present SP waveguides) within several propagation lengths. The proposed SP waveguide structure is much easier to fabricate as compared with some other types of SP waveguides for high integration.